Oh, I was confused regarding the implicit parentheses because of a piece of code I have which actually implements that formula above... Anyway, I now see why the code is implemented in the way it is :D
Anyway, that formula above is Bellman equation for the value function $V^\pi$ under policy $\pi$. Beautiful stuff.
i'd make a joke about financial math here, but wiki says: "In continuous-time optimization problems, the analogous equation is a partial differential equation which is usually called the Hamilton–Jacobi–Bellman equation."
Suppose I pick three vectors. Then the oriented volume of the parallelepiped formed by these vectors is is the determinant of the matrix with those vectors as solumns.
Reading at Mathworld, I came across the subject of tetrahedrons. Particularly calculating the volume with four known vertices. There's a formula which uses the triple product to calculate the volume of a parallelepiped. I'm very aware of why the triple product represents the volume of a parallele...
Suppose $L_2 = L_2([0,1],\lambda)$ over $\mathbb{R}$, where $\lambda$ denotes the Lebesgue measure. Let $T : L_2 \rightarrow L_2$ given by $$Tf(t) = \int_{0}^t f(s) ds$$. Compute formula for the adjoint and find the norm of the adjoint.
okay @Daminark ?
Here is how we do it. We know that $(f,T^*g) = (Tf,g)$ so let us compute $(Tf,g)$
@Semi at the notation, I had a root canal when I was 7, and later had a post stuck in there, it's just both of those guys did bad jobs so this guy is fixing it
Here's why I prefer having complex conjugation be on the first argument: When you do matrix multiplication, you typically write that as $v^Tv$ not $vv^T$
Let $G$ be a finite group of order $p^em$ with $p\not| m$, must every subgroup of $G$ which is a $p$-group be contained in a $p$-Sylow subgroup of $G$?
and if you wanted to take the norm of a complex vector, you'd do that as $v^\dagger v$. so it just seems more natural to put complex conjugation on the left.
@Daminark Suppose that $\|f\| = 1$ we have to get some bound on $\|Tf\|^2$ do you think it is better to use the norm from hilbert space or $L_2$ norm for computations ?
@Adeek total overkill approach: embedding $L^2[0,1]$ into $L^2(\Bbb R)$ isometrically by continuation with $0$, your operator is given by convolution with a constant function $1$, so you get a bound by Young's inequality $p=r=2, q= 1$
@MatheinBoulomenos here is good approach Suppose that $\|f\| = 1$. We compute $\|Tf\|^2 = \int_{0}^1 |Tf(t)|^2 dt = \int_{0}^1 1 * (\int_{0}^t f(t) dt)^2 dt$
I have a finite group $G$ of order $p^em$ with $p\not| m$ and subgroup of $G$ which is a $p$-group, must it be contained in a $p$-Sylow subgroup of $G$?
I likely won't finish in the recommended 6 semesters because I'm taking too much courses that interest me but I can't get credit for (because I have to take stupid stuff like numerical analysis, probability&statistics, programming projects ...)
I found a cool master's program that specializes in logic and set theory but also gives the opportunity to take a lot of geometry courses which is very tempting
@Alessandro: The trouble with logic and set theory is that — at least in the US — there are relatively few people interested in it and so jobs are almost impossible to get.
@TedShifrin If I'm good enough (which I doubt) I'd like to be employed as a pure mathematician. I doubt computing QR decompositions is going to help me with that
There is a majestic paper my Mac Lane
MacLane, Saunders. "Possible programs for categorists." Category Theory, Homology Theory and their Applications I. Springer, Berlin, Heidelberg, 1969. 123-131.
whose opening line is one of the most beautiful I've ever read:
Communication among Mathe...
I think this answer is bad, and propagates bad philosophy of math
It tells nobody anything about why anybody cares or should care about category theory
Arun Debray's answer is, in comparison, a lot more interesting
With the little perspective I have, I have to agree with something someone told me: one should study category theory once one knows a load of examples of concepts of category theory. It makes no sense to learn what a projective object is in a category, or what a terminal object is, or what a universal property is, or a kernel or an equalizer if one hasn't bumped into the obvious examples of such things in the practice.
It's been many years (like 43), but I recall that the first day of our grad alg top course was partly categories and functors. I mean, that sort of is the point of alg top.
So, maybe there's a counterargument to be made here that (a group of) geometers also have a certain preference for completely pictorial and physically motivated mathematics that is hard pushing for the algebraic and symbolic-minded people out there.
And he was like, not to self-promote but try my book, I think once you master that stuff, if you still like Riemann surfaces, you'd be ready for sheaves
But yeah I dunno, I think the main problem becomes when people start trying to extrapolate from "my taste in math is ___ on the picture/formal spectrum" to saying that this is how it ought be for everyone
Demonark, I constantly try to remind you all and future teachers in general that one has to remember that the average math student doesn't think/learn the same way you do.
True, though in principle might it balance out with having a few geometric types and a few algebraic types teaching? Like, get an idea of each and then zoom in however you'd like?
I think it's true that there's a fair proportion of students who are algebraically motivated and think naturally in terms of algebra (as in, intuit based on the mathematical symbols/rules/games? I don't really know how algebraists think), though, right?
literally all the analysts ive ever met have shown a clear preference for pictorial thinking but that may be because trying to do hard analysis in terms of formal sorts of manipulations and stuff is actually a total nightmare
I like to think that I think in terms of "structures". Not sure if everything I do is really formal. I sometimes just say to myself "these two must be isomorphic because of my gut feeling" or "this has to be natural in both arguments, becaue it would be really strange if it wasn't" or "I defined two maps that are both so canonical and I used so little that have to agree"
I have in mind one student who very much doesn't like algebra and isn't into geometry, and another who straight up dislikes anything that isn't analysis
And those two are the ones who like, I mean yeah sure go on with your pictures, but will never accept it fully. At some point or another everything has to be justified by something written
There is a majestic paper my Mac Lane
MacLane, Saunders. "Possible programs for categorists." Category Theory, Homology Theory and their Applications I. Springer, Berlin, Heidelberg, 1969. 123-131.
whose opening line is one of the most beautiful I've ever read:
Communication among Mathe...
